Mimetic Cosmology. Alexander Vikman. New Perspectives on Cosmology Institute of Physics of the Czech Academy of Sciences

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1 New Perspectives on Cosmology Mimetic Cosmology Alexander Vikman Institute of Physics of the Czech Academy of Sciences

2 This talk is mostly based on arxiv: , K. Hammer, A. Vikman arxiv: , JCAP 1506 (2015) 06, 028 L. Mirzagholi, A. Vikman arxiv: , JCAP 1406 (2014) 017 A. H. Chamseddine and V. Mukhanov, A. Vikman arxiv: , JCAP 1005 (2010) 012 I. Sawicki, E. Lim, A. Vikman

3 Plan of the talk

4 Plan of the talk Introduction

5 Plan of the talk Introduction What is mimetic construction?

6 Plan of the talk Introduction What is mimetic construction? Mimetic DM

7 Plan of the talk Introduction What is mimetic construction? Mimetic DM Imperfect DM with one scalar field and vorticity

8 Plan of the talk Introduction What is mimetic construction? Mimetic DM Imperfect DM with one scalar field and vorticity Mimetic Inflation

9 Plan of the talk Introduction What is mimetic construction? Mimetic DM Imperfect DM with one scalar field and vorticity Mimetic Inflation Open questions and conclusions

11 SM 5%

12 SM DM 5% 27%

13 SM DM 5% DE 27% 68%

14 SM DM 5% DE 27% 68% we feel them through gravity only!

15 Mimetic Matter Chamseddine, Mukhanov (2013)

16 Mimetic Matter Chamseddine, Mukhanov (2013) One can encode the conformal part of the physical metric in a scalar field:

17 Mimetic Matter Chamseddine, Mukhanov (2013) One can encode the conformal part of the physical metric in a scalar field: g µ = g

18 Mimetic Matter Chamseddine, Mukhanov (2013) One can encode the conformal part of the physical metric in a scalar field: g µ = g physical metric of free fall

19 Mimetic Matter Chamseddine, Mukhanov (2013) One can encode the conformal part of the physical metric in a scalar field: g µ = g physical metric of free fall auxiliary metric, dynamical variable

20 Mimetic Matter Chamseddine, Mukhanov (2013) One can encode the conformal part of the physical metric in a scalar field: g µ = g physical metric of free fall auxiliary metric, dynamical variable S [ g µ,, m] = Z d 4 x apple p 1 g 2 R (g)+l (g, m) g µ =g µ ( g, ) matter

21 Mimetic Matter Chamseddine, Mukhanov (2013)

22 Mimetic Matter Chamseddine, Mukhanov (2013) g µ = g physical metric of free fall auxiliary metric, dynamical variable

23 Mimetic Matter Chamseddine, Mukhanov (2013) g µ = g physical metric of free fall auxiliary metric, dynamical variable The theory becomes invariant with respect to Weyl transformations: g µ! 2 (x) g µ

24 Mimetic Matter Chamseddine, Mukhanov (2013) g µ = g physical metric of free fall auxiliary metric, dynamical variable The theory becomes invariant with respect to Weyl transformations: g µ! 2 (x) g µ The scalar field obeys the relativistic Hamilton-Jacobi equation: g =1

25 Mimetic Matter Chamseddine, Mukhanov (2013) g µ = g physical metric of free fall auxiliary metric, dynamical variable The theory becomes invariant with respect to Weyl transformations: g µ! 2 (x) g µ The scalar field obeys the relativistic Hamilton-Jacobi equation: g =1 g µ = g 1

26 the Hamilton-Jacobi equation g =1

27 the Hamilton-Jacobi equation g =1 corresponding four-velocity u µ µ is geodesic

28 the Hamilton-Jacobi equation g =1 corresponding four-velocity u µ µ is geodesic a µ = u r u µ = r (r r µ )= 1 µ (@ ) 2 =0

29 Modification of the Einstein equation

30 Modification of the Einstein equation S g µ

31 Modification of the Einstein equation S g µ G µ (g) T µ (g) (G (g) T =0

32 Modification of the Einstein equation S g µ G µ (g) T µ (g) (G (g) T =0 c.f. Einstein equations with dust or DM: G µ = T µ + u µ u = G T

33 Modification of the Einstein equation S g µ G µ (g) T µ (g) (G (g) T =0 c.f. Einstein equations with dust or DM: G µ = T µ + u µ u = G T

34 Modification of the Einstein equation S g µ G µ (g) T µ (g) (G (g) T =0 c.f. Einstein equations with dust or DM: G µ = T µ + u µ u = G T Dark Matter energy density: 4-velocity: u µ µ

35 S [ g µ,, m] = Z d 4 x apple p 1 g 2 R (g)+l (g, m) g µ =g µ ( g, ) with g µ ( g, )= g

36 S [ g µ,, m] = Z d 4 x apple p 1 g 2 R (g)+l (g, m) g µ =g µ ( g, ) with g µ ( g, )= g S 0 [ g, '] = Z d 4 x p g XR( g)+ 3 2 g X, X, X L m ( g,, m), where X = 1 2 g,,

37 S [ g µ,, m] = Z d 4 x apple p 1 g 2 R (g)+l (g, m) g µ =g µ ( g, ) with g µ ( g, )= g S 0 [ g, '] = Z d 4 x p g XR( g)+ 3 2 g X, X, X Brans-Dicke! = 3/2 L m ( g,, m), where X = 1 2 g,,

38 S [ g µ,, m] = Z d 4 x apple p 1 g 2 R (g)+l (g, m) g µ =g µ ( g, ) with g µ ( g, )= g S 0 [ g, '] = Z d 4 x p g XR( g)+ 3 2 g X, X, X Brans-Dicke! = 3/2 L m ( g,, m), where X = 1 2 g,, is not in the Horndeski (1974) construction of the most general scalar-tensor theory with second order equations of motion

S [ g µ,, m] = Z d 4 x apple p 1 g 2 R (g)+l (g, m) g µ =g µ ( g, ) with g µ ( g, )= g µ g @ @ S 0 [ g, '] = Z d 4 x p g XR( g)+ 3 2 g X, X, X Brans-Dicke!

39 S [ g µ,, m] = Z d 4 x apple p 1 g 2 R (g)+l (g, m) g µ =g µ ( g, ) with g µ ( g, )= g S 0 [ g, '] = Z d 4 x p g XR( g)+ 3 2 g X, X, X Brans-Dicke! = 3/2 L m ( g,, m), where X = 1 2 g,, is not in the Horndeski (1974) construction of the most general scalar-tensor theory with second order equations of motion But it is still a system with one degree of freedom + standard two polarizations for the graviton!

40 Lagrange Multiplier and Weyl-Invariance Hammer, Vikman (2015)

41 Lagrange Multiplier and Weyl-Invariance S 0 [ g, ', X, ]= Z d 4 x p g apple XR( g)+ 3 2 g X, X X + Hammer, Vikman (2015) X 1 2 g,,

42 Lagrange Multiplier and Weyl-Invariance S 0 [ g, ', X, ]= Z d 4 x p g apple XR( g)+ 3 2 g X, X X + Hammer, Vikman (2015) X 1 2 g,, Weyl-invariance: g µ! 2 (x) g µ,!, X! 2 (x) X,! 2 (x),

43 Lagrange Multiplier and Weyl-Invariance S 0 [ g, ', X, ]= Z d 4 x p g apple XR( g)+ 3 2 g X, X X + Hammer, Vikman (2015) X 1 2 g,, Weyl-invariance: g µ! 2 (x) g µ,!, X! 2 (x) X,! 2 (x), gauge invariant variables g µ =(2X) 1 g µ, ' = ', X = X, =2X,

44 Lagrange Multiplier and Weyl-Invariance S 0 [ g, ', X, ]= Z d 4 x p g apple XR( g)+ 3 2 g X, X X + Hammer, Vikman (2015) X 1 2 g,, Weyl-invariance: g µ! 2 (x) g µ,!, X! 2 (x) X,! 2 (x), gauge invariant variables g µ =(2X) 1 g µ, ' = ', X = X, =2X, S 0 [g, ', ] = Z d 4 x p g apple 1 2 R (g)+ 2 g ', ', 1.

45 Mimetic Dark Matter via Lagrange Multiplier Chamseddine, Mukhanov; Golovnev; Barvinsky (2013) Lim, Sawicki, Vikman (2010);

46 Mimetic Dark Matter via Lagrange Multiplier Chamseddine, Mukhanov; Golovnev; Barvinsky (2013) Lim, Sawicki, Vikman (2010); constraint via Lagrange multiplier Z S [g,,, SM] = d 4 x p g (@ ) R (@ ) L SM

47 Mimetic Dark Matter via Lagrange Multiplier Chamseddine, Mukhanov; Golovnev; Barvinsky (2013) Lim, Sawicki, Vikman (2010); constraint via Lagrange multiplier Z S [g,,, SM] = d 4 x p g (@ ) R (@ ) L SM T µ = u µ u

48 Mimetic Dark Matter via Lagrange Multiplier Chamseddine, Mukhanov; Golovnev; Barvinsky (2013) Lim, Sawicki, Vikman (2010); constraint via Lagrange multiplier Z S [g,,, SM] = d 4 x p g (@ ) R (@ ) L SM T µ = u µ u Dark Matter Lagrange multiplier is the energy density u µ µ

49 dust / DM via Lagrange multiplier (@ ) 2 1

50 dust / DM via Lagrange multiplier (@ ) 2 1 Cosmological Constant / DE via Lagrange multiplier (r µ V µ 1) M. Henneaux and C. Teitelboim (1989)

51 Disformal Transformation Nathalie Deruelle and Josephine Rua (2014), Domènech et al. (2015) One obtains the same dynamics (the same Einstein equations), if instead of varying the Einstein-Hilbert action with respect to the metric g µ one plugs in a disformal transformation (Bekenstein 1993) g µ = F (,w) `µ + H (,w)@ w = w 2 with and H + w 6= 0 and varies with respect to `µ,

52 Disformal Transformation Nathalie Deruelle and Josephine Rua (2014), Domènech et al. (2015) One obtains the same dynamics (the same Einstein equations), if instead of varying the Einstein-Hilbert action with respect to the metric g µ one plugs in a disformal transformation (Bekenstein 1993) g µ = F (,w) `µ + H (,w)@ w = w 2 with and H + w 6= 0 and varies with respect to `µ, Mimetic gravity is an exception! And it does provide new dynamics!

53 Mimicking any cosmological evolution Chamseddine, Mukhanov, Vikman (2013) Lim, Sawicki, Vikman; (2010)

54 Mimicking any cosmological evolution Just add a potential V ( )! Chamseddine, Mukhanov, Vikman (2013) Lim, Sawicki, Vikman; (2010)

55 Mimicking any cosmological evolution Just add a potential V ( )! Chamseddine, Mukhanov, Vikman (2013) Lim, Sawicki, Vikman; (2010) T µ = u µ u + g µ V

56 Mimicking any cosmological evolution Just add a potential V ( )! Chamseddine, Mukhanov, Vikman (2013) Lim, Sawicki, Vikman; (2010) T µ = u µ u + g µ V g =1 Convenient to take as time

57 Mimicking any cosmological evolution Just add a potential V ( )! Chamseddine, Mukhanov, Vikman (2013) Lim, Sawicki, Vikman; (2010) T µ = u µ u + g µ V g =1 Convenient to take as time potential provides time-dependent pressure

58 Mimicking any cosmological evolution Just add a potential V ( )! Chamseddine, Mukhanov, Vikman (2013) Lim, Sawicki, Vikman; (2010) T µ = u µ u + g µ V g =1 Convenient to take as time potential provides time-dependent pressure Enough freedom to obtain any cosmological evolution!

59 In particular inflation as V ( )= m2 2 m 4 2 e +1 gives the same cosmological potential in the standard case

60 Perturbations I Chamseddine, Mukhanov, Vikman (2013) Lim, Sawicki, Vikman; (2010) Even with potential, the energy still moves along the timelike geodesics

61 Perturbations I Chamseddine, Mukhanov, Vikman (2013) Lim, Sawicki, Vikman; (2010) Even with potential, the energy still moves along the timelike geodesics c S =0

62 Perturbations I Chamseddine, Mukhanov, Vikman (2013) Lim, Sawicki, Vikman; (2010) Even with potential, the energy still moves along the timelike geodesics c S =0 Newtonian potential: Z H =C 1 (x) 1 adt + H a C 2 (x) a Here on all scales but in the usual cosmology it is an approximation for superhorizon scales

63 Next term in the gradient expansion Chamseddine, Mukhanov, Vikman (2013)

64 Next term in the gradient expansion Chamseddine, Mukhanov, Vikman (2013) ( ) 2 the unique quadratic term with higher derivatives

65 Next term in the gradient expansion Chamseddine, Mukhanov, Vikman (2013) ( ) 2 the unique quadratic term with higher derivatives Z r µ r r µ r is not that useful: d 4 x p Z g ;µ; ;µ; = d 4 x p g ( ) 2 R µ ;µ ;

66 Next term in the gradient expansion Chamseddine, Mukhanov, Vikman (2013) ( ) 2 the unique quadratic term with higher derivatives There are no new degrees of freedom, because higher time derivatives can be eliminated by differentiating the Hamilton-Jacobi equation. Z r µ r r µ r is not that useful: d 4 x p Z g ;µ; ;µ; = d 4 x p g ( ) 2 R µ ;µ ;

67 Next term in the gradient expansion Chamseddine, Mukhanov, Vikman (2013) ( ) 2 the unique quadratic term with higher derivatives There are no new degrees of freedom, because higher time derivatives can be eliminated by differentiating the Hamilton-Jacobi equation. Z r µ r r µ r is not that useful: d 4 x p Z g ;µ; ;µ; = d 4 x p g ( ) 2 R µ ;µ ; expansion = = r µ u µ

68 CHARGE CONSERVATION

69 CHARGE CONSERVATION no potential! + c symmetry

70 CHARGE CONSERVATION no potential! + c symmetry r µ J µ =0 Noether current:

71 CHARGE CONSERVATION no potential! + c symmetry r µ J µ =0 Noether current: charge density n / a 3

72 Imperfection

73 Imperfection J µ shift-charge current Eckart frame

74 Imperfection J µ shift-charge current Energy flow: timelike eigenvector of energy-momentum tensor Eckart frame Landau-Lifshitz frame

75 Imperfection / 2 J µ shift-charge current Energy flow: timelike eigenvector of energy-momentum tensor Eckart frame Landau-Lifshitz frame

76 Imperfection in the Noether current J µ = u µ,µ

77 Imperfection in the Noether current J µ = u µ,µ expansion = = r µ u µ

78 Imperfect Dark Matter Mirzagholi, Vikman (2014)

79 Imperfect Dark Matter (no potential) Mirzagholi, Vikman (2014) T µ = "u µ u p? µ +q µ u + q u µ

80 Imperfect Dark Matter (no potential) Mirzagholi, Vikman (2014) T µ = "u µ u p? µ +q µ u + q u µ? µ = g µ u µ u expansion = r µ u µ

81 Imperfect Dark Matter (no potential) Mirzagholi, Vikman (2014) T µ = "u µ u p? µ +q µ u + q u µ energy flow q µ =? µ r? µ = g µ u µ u expansion = r µ u µ

82 Imperfect Dark Matter (no potential) Mirzagholi, Vikman (2014) T µ = "u µ u p? µ +q µ u + q u µ energy flow q µ =? µ r energy density " = 1 2 2? µ = g µ u µ u expansion = r µ u µ

83 Imperfect Dark Matter (no potential) Mirzagholi, Vikman (2014) T µ = "u µ u p? µ +q µ u + q u µ energy flow q µ =? µ r energy density " = pressure p = ? µ = g µ u µ u expansion = r µ u µ

84 Imperfect Dark Matter (no potential) Mirzagholi, Vikman (2014) T µ = "u µ u p? µ +q µ u + q u µ energy flow q µ =? µ r energy density " = pressure p = ? µ = g µ u µ u expansion = r µ u µ = u µ r µ

85 Vorticity for a single dof DM!

86 Vorticity for a single dof DM! in the frame moving with the charges (Eckart frame) V µ = J µ p J J

87 Vorticity for a single dof DM! in the frame moving with the charges (Eckart frame) V µ = J µ p J J Vorticity vector: µ (V )= 1 2 " µ V V ; ' 2 2 " µ,,,

88 Vorticity for a single dof DM! in the frame moving with the charges (Eckart frame) V µ = J µ p J J = = r µ u µ Vorticity vector: µ (V )= 1 2 " µ V V ; ' 2 2 " µ,,,

89 Vorticity for a single dof DM! in the frame moving with the charges (Eckart frame) V µ = J µ p J J = = r µ u µ Vorticity vector: µ (V )= 1 2 " µ V V ; ' 2 2 " µ,,, the circulation is conserved up to O 2

90 Perturbations II

91 Perturbations II + H c 2 s a 2 + Ḣ =0 with the sound speed c 2 s = 2 3

92 Perturbations II + H c 2 s a 2 + Ḣ =0 with the sound speed c 2 s = 2 3 Newtonian potential: =

93 Perturbations II + H c 2 s a 2 + Ḣ =0 with the sound speed c 2 s = 2 3 Ramazanov,Capela (2014) Newtonian potential: = c S 10 5

94 Background cosmology

95 Background cosmology " = n +3c2 S ext p =3c 2 SP ext

96 Background cosmology " = n +3c2 S ext p =3c 2 SP ext DM n / a 3

97 Background cosmology " = n +3c2 S ext p =3c 2 SP ext DM n / a 3 G e = G N 1+3c 2 S

98 Background cosmology " = n +3c2 S ext p =3c 2 SP ext DM n / a 3 G e = G N 1+3c 2 S G N bounds are mild: 3 c 2 S matter c 2 S radiation ± Narimani, Scott, Afshordi(2014)

99 mimetic construction and inflation n / a 3

100 mimetic construction and inflation n / a 3 shift-symmetry breaking is needed for mimetic DM!

101

102 Generating shift-charge (DM) during radiation domination époque

103 Generating shift-charge (DM) during radiation domination époque t[7]= ( ) H 1

104 Generating shift-charge (DM) during radiation domination époque t[7]= ( ) r µ J µ = ( ) 2 H 1

105 Generating shift-charge (DM) during radiation domination époque t[7]= ( ) r µ J µ = ( ) 2 H 1 =3H

106 Generating shift-charge (DM) during radiation domination époque t[7]= ( ) r µ J µ = ( ) 2 H 1 n (t cr ) / a 3 Z =3H dt 0 a 3 H 2 ' rad (t cr )

107 Generating shift-charge (DM) during radiation domination époque t[7]= ( ) r µ J µ = ( ) 2 H 1 n (t cr ) / a 3 Z =3H dt 0 a 3 H 2 ' rad (t cr ) at DM / radiation equality DM (z eq )= rad (z eq )

108 Generating shift-charge (DM) during radiation domination époque t[7]= ( ) r µ J µ = ( ) 2 H 1 n (t cr ) / a 3 Z =3H dt 0 a 3 H 2 ' rad (t cr ) at DM / radiation equality DM (z eq )= rad (z eq ) ' acr a eq ' z eq z cr

109 Generating shift-charge (DM) during radiation domination époque t[7]= ( ) r µ J µ = ( ) 2 H 1 n (t cr ) / a 3 Z =3H dt 0 a 3 H 2 ' rad (t cr ) at DM / radiation equality DM (z eq )= rad (z eq ) acr ' a eq ' z eq z cr apple

110 Generating shift-charge (DM) during radiation domination époque t[7]= ( ) r µ J µ = ( ) 2 H 1 n (t cr ) / a 3 Z =3H dt 0 a 3 H 2 ' rad (t cr ) at DM / radiation equality DM (z eq )= rad (z eq ) acr ' a eq ' z eq z cr apple apple 10 10

111 Generating shift-charge (DM) during radiation domination époque t[7]= ( ) r µ J µ = ( ) 2 H 1 n (t cr ) / a 3 Z =3H dt 0 a 3 H 2 ' rad (t cr ) at DM / radiation equality DM (z eq )= rad (z eq ) acr ' a eq ' z eq z cr apple apple T cr ' T eq & 10 GeV

112 Back to inflation with potential and ( ) 2

113 Quantization

114 Quantization the action Z S = 1 2 d d 3 x c 2 s

115 Quantization the action Z S = 1 2 d d 3 x c 2 s short wavelength quantum fluctuations r cs ~ k 3/2 k ' match with the long-wave-length limit

116 Perturbations in Mimetic Inflation

117 Perturbations in Mimetic Inflation on scale ' k 1

118 Perturbations in Mimetic Inflation on scale ' k 1 Newtonian potential ' p c s / H cs k'h

119 Perturbations in Mimetic Inflation on scale ' k 1 Newtonian potential ' p c s / H cs k'h for c S 1 ' c 1/2 H S cs k'h

120 Perturbations in Mimetic Inflation on scale ' k 1 Newtonian potential ' p c s / H cs k'h for c S 1 ' c 1/2 H S cs k'h for c S 1 ' c 1/2 S H c S k'h

121 Perturbations in Mimetic Inflation on scale ' k 1 Newtonian potential ' p c s / H cs k'h for c S 1 ' c 1/2 H S cs k'h for c S 1 ' c 1/2 Gravitational Waves are unchanged h ' H k'h S H c S k'h

122 Perturbations in Mimetic Inflation on scale ' k 1 Newtonian potential ' p c s / H cs k'h for c S 1 ' c 1/2 H S cs k'h for c S 1 ' c 1/2 Gravitational Waves are unchanged h ' H k'h S H c S k'h spectral indices n S 1=n T

123 Perturbations in Mimetic Inflation on scale ' k 1 Newtonian potential ' p c s / H cs k'h for c S 1 ' c 1/2 H S cs k'h for c S 1 ' c 1/2 Gravitational Waves are unchanged h ' H k'h S H c S k'h spectral indices n S 1=n T small sound speed h But the non-gaussianity is vanishingly small!

124 Further Directions

125 Further Directions Caustics?

126 Further Directions Caustics? Galactic Halos?

127 Further Directions Caustics? Galactic Halos? Gravitational collapse?

128 Further Directions Caustics? Galactic Halos? Gravitational collapse? Accretion of the mimetic DM?

129 Further Directions Caustics? Galactic Halos? Gravitational collapse? Accretion of the mimetic DM? Bullet Cluster?

130 Further Directions Caustics? Galactic Halos? Gravitational collapse? Accretion of the mimetic DM? Bullet Cluster? Quantum corrections?

131 Conclusions

132 Conclusions New large class of Weyl-invariant scalar-tensor theories

133 Conclusions New large class of Weyl-invariant scalar-tensor theories Unification of Dark Matter with Dark Energy in one degree of freedom.

134 Conclusions New large class of Weyl-invariant scalar-tensor theories Unification of Dark Matter with Dark Energy in one degree of freedom. Imperfect Dark Matter, with a small sound speed and vorticity. Only one free parameter for the late Universe.

135 Conclusions New large class of Weyl-invariant scalar-tensor theories Unification of Dark Matter with Dark Energy in one degree of freedom. Imperfect Dark Matter, with a small sound speed and vorticity. Only one free parameter for the late Universe. New class of inflationary models with suppressed gravity waves and low non-gaussianity.

136 Conclusions New large class of Weyl-invariant scalar-tensor theories Unification of Dark Matter with Dark Energy in one degree of freedom. Imperfect Dark Matter, with a small sound speed and vorticity. Only one free parameter for the late Universe. New class of inflationary models with suppressed gravity waves and low non-gaussianity.!anks a lot for a"ention!

Mimetic dark matter. The mimetic DM is of gravitational origin. Consider a conformal transformation of the type:

Mimetic dark matter. The mimetic DM is of gravitational origin. Consider a conformal transformation of the type: Mimetic gravity Frederico Arroja FA, N. Bartolo, P. Karmakar and S. Matarrese, JCAP 1509 (2015) 051 [arxiv:1506.08575 [gr-qc]] and JCAP 1604 (2016) no.04, 042 [arxiv:1512.09374 [gr-qc]]; S. Ramazanov,